1. INTRODUCTION

How does the brain work? What kind of language, what concepts, what terms are needed to relate in one direction to what the brain is found to accomplish and in the other to the underlying pattern of neural activity? This paper proffers an answer to that question with respect to the perception of counterphase flicker. The mechanics of counterphase flicker can be simply described by analogy with the radio reception technology of the first half of the last century. Moreover, it is plausible that the critical function in counterphase flicker is realized in the retina, so that the same analogy tells us how to model the behavior of retinal ganglion cells.

2. COUNTERPHASE FLICKER

There are two identical visual stimuli, identical except that the illumination in one is modulated in exact antiphase to the other. One stimulus is seen through one eye, the other through the other eye [Fig. 1(a)]. Cavonius [1], for example, presented two 1 deg circular patches of light haploscopically in Maxwellian view. The two stimuli appeared as a single image: “It is difficult not to fuse such stimuli” ( [1], p. 274). He measured thresholds over the frequency range of 0.1 to 50 Hz. Figure 1 illustrates the problem. If the two illuminances are superposed, the peaks and troughs cancel, and there is no temporal modulation [Fig. 1(b)]. It follows that counterphase flicker is perceptible by virtue of a nonlinearity between each retina and the point of binocular convergence. What is that nonlinearity? How might it most simply be modeled?

 

Fig. 1. (a) Counterphase stimuli, showing that (b) linear superposition precludes perception.

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Figure 2 proffers a simple solution. In Fig. 2(b), the sinusoidal inputs are first differentiated, then half-wave rectified in Fig. 2(c), and finally superposed in Fig. 2(d). Working backward, there has to be a half-wave rectification; otherwise, the troughs in one input would cancel the peaks in the other. There also has to be a prior differentiation so that the half-wave rectification exactly eliminates the negative excursions of the modulation [dotted curves in Fig. 2(b)].

 

Fig. 2. Counterphase stimuli (again), showing the effect of (b) differentiation, (c) half-wave rectification, and (d) superposition. The negative excursions in (b) are shown dotted.

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The next question is: Where in the visual pathway does this take place? That is a matter of conjecture, but note that retinal ganglion cells receive both positive (excitatory) and negative (inhibitory) inputs, but transmit action potentials of one polarity only (half-wave rectification).

There is one serious defect in the foregoing argument. Figure 2(b) envisages negative excursions (dotted curves) in the mathematical differentiation of the sinusoidal modulations. How are those negative excursions realized? A reworking of the argument (Fig. 3) provides an answer.

 

Fig. 3. (a) Counterphase stimuli modulating positive and negative Poisson processes, (b) in a background of Gaussian noise, (c) after half-wave rectification, and (d) after superposition.

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As a preliminary, Laming ( [2], Chaps. 5 and 6) has proposed a simple model of retinal ganglion cell function for the purpose of calculating threshold values, psychometric functions, and signal detection operating characteristics. Under uniform illumination, the positive and negative inputs to the cell both absorb Poisson processes of quantal absorptions. These two input streams are statistically independent and are assumed to be balanced. This means that the Poisson means, positive and negative, cancel, while the statistical fluctuations combine in square measure to generate an approximately Gaussian noise process of power proportional to the illuminance. Write the Gaussian density as

In Fig. 3(a), the Poisson densities are modulated sinusoidally, the right eye in antiphase to the left. When the two inputs are combined [Fig. 3(b)], the sinusoidal modulations do not entirely cancel, because the negative input is delayed (by an interval) relative to the positive. Hughes and Maffei [3] examined the retinal ganglion cell response to flicker in the cat and reported phase shifts at different frequencies equating to a delay of approximately 65 ms. In consequence, a frequency-dependent proportion, $|\phi ({{2}}\pi f)|$, of the temporal modulation is preserved, where $\phi ()$ is a temporal transfer function. The Gaussian noise background is now combined with a mean varying as ${\rm{LC}}|\phi ({{2}}\pi f)|\;{\cos}\;({{2}}\pi ft)$. For simplicity, write the transmitted process in Fig. 3(b) as

In the interests of clarity, the illustrative diagrams in Fig. 3 have been drawn with an exaggerated signal-to-noise ratio. For smaller depths of modulation, expansion in a Maclaurin series gives, for small contrast $C$,

This interlaced output is seen as (counterphase) flicker of frequency ${{2}}f$.

3. APPARENT FREQUENCY OF COUNTERPHASE FLICKER

Cavonius et al. [4] asked observers to adjust the frequency and amplitude of an in-phase stimulus, viewed binocularly, to match a neighboring counterphase field, modulated at about 2.5 times threshold. Observers set the in-phase frequency to twice the counterphase frequency, and the amplitude to a value that, for a given frequency, was a constant ratio of the modulation of the counterphase field.

“As long as the depth of modulation was not greater than about ${{4}} \times$ the detection threshold, counterphase flicker could be completely matched by in-phase flicker at twice the frequency: under these conditions the two fields are indistinguishable” ( [4], pp. 154–5).

The second harmonic signal, which is the subject of the match, is plain to see in Fig. 3(d). The “2.5 times its threshold” is important. Since it is the frequency that is to be matched, a level is needed suitably above threshold. At different depths of modulation below the “${{4}} \times$ the detection threshold” limit, the matching in-phase amplitude was a constant proportion of the counterphase amplitude ( [4], Fig. 2), although the ratio of the two varied greatly with frequency. On this basis, Cavonius et al. ( [4], p. 155) proposed that “counterphase stimulation cancels the fundamental component of the neural signal and the sensation of flicker is produced by higher even-harmonics,” in which case the nonlinearity producing the second harmonic must be of a “linear-rectifier type” [as in Fig. 2(b)]. At higher depths of modulation higher-order, however, even harmonics become detectable.

“... observers were unable to tell whether flicker was in-phase or counterphase; but that with strong modulation, counterphase flicker had a qualitatively different appearance: observers usually described it as “rough,” as if it comprised more than one frequency. At those higher depths of modulation, the fields are qualitatively different, and no match is possible” ( [4], pp. 153–4).

Cavonius et al. endeavored to cancel the internal second harmonic by adding a second harmonic component to the stimulus. If the component that is perceived to have twice the fundamental frequency is the second harmonic of the physical stimulus, then adding a second harmonic component to that stimulus should modulate the perceived counterphase flicker. Of course, if the added second harmonic were of sufficient strength, it would be detectable in its own right, so what is needed, to avoid that artifact, is a phase-sensitive interaction. But “if a near-threshold second harmonic is added to the counterphase stimulus and its phase rotated through 360 deg, the perceived flicker never waxes or wanes” ( [4], p. 155). Figures 2(d) and 3(d) show that the variation at frequency ${{2}}f$, perceived as counterphase flicker, is derived from the interlacing of counterphase modulations at the fundamental frequency and has nothing to do with the second harmonic of the stimulus waveform.

4. DISCUSSION

Figure 1 shows that the perceptibility of counterphase flicker means that there is a nonlinearity between each retina and binocular fusion. This points to that part of the visual pathway up to and including the simple cells located in layer IVb of the striate cortex, as those cells are predominantly monocular [5–9], but the complex cells beyond are not. Although the lateral geniculate nucleus receives inputs from both eyes, those inputs appear to be transmitted independently through both layers IVc (circularly symmetric units) and IVb (simple cells) before any significant interaction takes place.

Within that part of the visual pathway, a plausible locus of the nonlinearity is the retinal ganglion cells. Laming [10] has applied the model in Fig. 3 to “auditory” thresholds of retinal ganglion cells in the cat measured by Barlow and Levick [11] and Barlow, Fitzhugh, and Kuffler [12]. “Auditory” means a threshold adjustment of the stimulus luminance “so as to produce the weakest reliably detectable perturbation of the maintained discharge as played over a loudspeaker” ( [11], p. 739). Summarizing their results as of 1969, Barlow and Levick wrote “It is as if the ganglion cells at adapting levels above about ${{10}^{- 2}}\;{\rm{cd}}/{{\rm{m}}^2}$ … acquired a differential input” ( [13], p. 716). If the model in Fig. 3 adequately represents the function of retinal ganglion cells, then transmission through the ganglion cell layer would give effect to the nonlinearity needed to make counterphase flicker perceptible.

In Figs. 2 and 3, I have chosen to be simple rather than exact. For example, the same model is proposed for all retinal ganglion cells, even though they vary greatly. Moreover, counterphase modulation at frequency ${{2}}f$ will still be generated in Fig. 3(d), even though some small element of the physical stimulus is transmitted through the differential coupling in Fig. 3(b), but it has to be small. A limited failure of differential coupling would displace the half-wave rectification from the mean of the modulation waveform and produce uneven peaks and troughs.

The mechanics of the perception of counterphase flicker are most simply described in terms of the radio reception technology of the first half of the last century. The maintained discharge from a retinal ganglion cell is analogous to shot noise in a thermionic vacuum tube. A stimulus is perceived if it stands out sufficiently above that noise. Half-wave rectification and differentiation (capacitor coupling) are likewise reminiscent of vacuum tube technology; in Fig. 3(b), however, mathematical differentiation is replaced by differential coupling. The emphasis is on the detection and amplification of signals by noise-prone devices.

If electrophysiological observations are to be correctly interpreted, there must first be an overarching functional map of the visual pathway. The development here has been based on counterphase flicker, but it should not pass without notice that these same ideas have application to the modelling of thresholds, psychometric functions, and signal detection characteristics in all five sensory modalities [2,14–16] and, therefore, the early stages of sensory pathways in general.